108 is 36% of what number? write and solve a proportion to solve the problem.

Learning Objectives

By the cease of this section, yous will be able to:

• Use the definition of proportion
• Solve proportions
• Solve applications using proportions
• Write per centum equations as proportions
• Interpret and solve percent proportions

Be Prepared vi.5

Before you lot get started, take this readiness quiz.

1. Simplify: $\frac{\frac{\mathrm{i}}{3}}{\mathrm{four}}.$
   If you missed this problem, review Case 4.44.
2. Solve: $\frac{x}{\mathrm{iv}}=20.$
   If you missed this problem, review Case 4.99.
3. Write as a rate: Sale rode his bike $24$ miles in $\mathrm{ii}$ hours.
   If you missed this problem, review Case five.63.

Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. When 2 ratios or rates are equal, the equation relating them is chosen a proportion.

Proportion

A proportion is an equation of the form $\frac{a}{b}=\frac{c}{d},$ where $b\ne 0,d\ne 0.$

The proportion states two ratios or rates are equal. The proportion is read $\text{"}a$ is to $b,$ as $c$ is to $d\text{".}$

The equation $\frac{\mathrm{i}}{2}=\frac{4}{8}$ is a proportion considering the two fractions are equal. The proportion $\frac{1}{2}=\frac{\mathrm{iv}}{\mathrm{viii}}$ is read $\text{"}\mathrm{ane}$ is to $\mathrm{ii}$ as $4$ is to $8\text{".}$

If we compare quantities with units, we have to be certain we are comparing them in the right lodge. For case, in the proportion $\frac{\text{twenty students}}{\text{1 instructor}}=\frac{\text{lx students}}{\text{3 teachers}}$ we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

Case half dozen.xl

Write each sentence as a proportion:

1. ⓐ $3$ is to $7$ as $15$ is to $35.$
2. ⓑ $5$ hits in $\mathrm{viii}$ at bats is the aforementioned as $30$ hits in $48$ at-bats.
3. ⓒ $\text{\$1.fifty}$ for $\mathrm{vi}$ ounces is equivalent to $\text{\$2.25}$ for $\mathrm{ix}$ ounces.

Try It 6.79

Write each judgement every bit a proportion:

1. ⓐ $5$ is to $9$ as $20$ is to $36.$
2. ⓑ $7$ hits in $11$ at-bats is the same every bit $28$ hits in $44$ at-bats.
3. ⓒ $\text{\$two.50}$ for $\mathrm{viii}$ ounces is equivalent to $\text{\$3.75}$ for $12$ ounces.

Attempt It 6.80

Write each sentence as a proportion:

1. ⓐ $6$ is to $7$ as $36$ is to $42.$
2. ⓑ $8$ adults for $36$ children is the aforementioned every bit $12$ adults for $54$ children.
3. ⓒ $\text{\$3.75}$ for $6$ ounces is equivalent to $\text{\$2.50}$ for $\mathrm{four}$ ounces.

Look at the proportions $\frac{1}{2}=\frac{\mathrm{iv}}{8}$ and $\frac{2}{3}=\frac{6}{\mathrm{ix}}.$ From our work with equivalent fractions nosotros know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if information technology contains fractions with larger numbers?

To decide if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the reverse numerator (diagonally across the equal sign). The results are called a cross products because of the cross formed. The cantankerous products of a proportion are equal.

Cross Products of a Proportion

For any proportion of the form $\frac{a}{b}=\frac{c}{d},$ where $b\ne 0,d\ne 0,$ its cross products are equal.

Cantankerous products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are the equal, we have a proportion.

Case vi.41

Decide whether each equation is a proportion:

1. ⓐ $\frac{\mathrm{four}}{\mathrm{nine}}=\frac{12}{28}$
2. ⓑ $\frac{17.5}{\mathrm{37.v}}=\frac{7}{\mathrm{xv}}$

Try It 6.81

Decide whether each equation is a proportion:

1. ⓐ $\frac{7}{9}=\frac{54}{72}$
2. ⓑ $\frac{\mathrm{24.five}}{\mathrm{45.five}}=\frac{\mathrm{vii}}{13}$

Try It 6.82

Determine whether each equation is a proportion:

1. ⓐ $\frac{8}{\mathrm{nine}}=\frac{56}{73}$
2. ⓑ $\frac{28.5}{52.5}=\frac{8}{15}$

Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations notwithstanding apply. In the next example, we will solve a proportion by multiplying by the Least Mutual Denominator (LCD) using the Multiplication Belongings of Equality.

Instance half-dozen.42

Try It 6.83

Solve the proportion: $\frac{n}{84}=\frac{\mathrm{eleven}}{12}.$

Try It half-dozen.84

Solve the proportion: $\frac{y}{96}=\frac{13}{12}.$

When the variable is in a denominator, we'll use the fact that the cross products of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then ready them equal. Then we solve the resulting equation using our familiar techniques.

Example half dozen.43

Try It 6.85

Solve the proportion: $\frac{91}{b}=\frac{\mathrm{seven}}{\mathrm{v}}.$

Attempt It 6.86

Solve the proportion: $\frac{39}{c}=\frac{\mathrm{thirteen}}{\mathrm{viii}}.$

Example 6.44

Solve: $\frac{52}{91}=\frac{\mathrm{-4}}{y}.$

Try It 6.87

Solve the proportion: $\frac{84}{98}=\frac{\mathrm{-6}}{x}.$

Attempt It 6.88

Solve the proportion: $\frac{\mathrm{-7}}{y}=\frac{105}{135}.$

Solve Applications Using Proportions

The strategy for solving applications that we have used before in this chapter, also works for proportions, since proportions are equations. When we ready up the proportion, we must brand sure the units are correct—the units in the numerators friction match and the units in the denominators match.

Example 6.45

When pediatricians prescribe acetaminophen to children, they prescribe $\mathrm{v}$ milliliters (ml) of acetaminophen for every $25$ pounds of the kid'southward weight. If Zoe weighs $\mathrm{eighty}$ pounds, how many milliliters of acetaminophen will her md prescribe?

Try It six.89

Pediatricians prescribe $5$ milliliters (ml) of acetaminophen for every $25$ pounds of a child's weight. How many milliliters of acetaminophen will the dr. prescribe for Emilia, who weighs $60$ pounds?

Try It six.ninety

For every $1$ kilogram (kg) of a kid's weight, pediatricians prescribe $\mathrm{xv}$ milligrams (mg) of a fever reducer. If Isabella weighs $12$ kg, how many milligrams of the fever reducer will the pediatrician prescribe?

Instance 6.46

One brand of microwave popcorn has $120$ calories per serving. A whole bag of this popcorn has $3.5$ servings. How many calories are in a whole bag of this microwave popcorn?

Try It vi.91

Marissa loves the Caramel Macchiato at the coffee store. The $16$ oz. medium size has $240$ calories. How many calories will she get if she drinks the large $20$ oz. size?

Try Information technology 6.92

Yaneli loves Starburst candies, merely wants to go on her snacks to $100$ calories. If the candies accept $160$ calories for $\mathrm{viii}$ pieces, how many pieces can she have in her snack?

Example six.47

Josiah went to Mexico for spring interruption and changed $\text{\$325}$ dollars into Mexican pesos. At that time, the exchange rate had $\text{\$ane}$ U.S. is equal to $12.54$ Mexican pesos. How many Mexican pesos did he get for his trip?

Effort It half dozen.93

Yurianna is going to Europe and wants to change $\text{\$800}$ dollars into Euros. At the current exchange rate, $\text{\$1}$ US is equal to $0.738$ Euro. How many Euros will she have for her trip?

Endeavour It 6.94

Corey and Nicole are traveling to Japan and need to exchange $\text{\$600}$ into Japanese yen. If each dollar is $94.1$ yen, how many yen volition they get?

Write Percent Equations As Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percentage equations by using the proportion method. The proportion method for solving percent problems involves a per centum proportion. A percent proportion is an equation where a per centum is equal to an equivalent ratio.

For example, $\text{lx\%}=\frac{\mathrm{lx}}{100}$ and nosotros can simplify $\frac{60}{100}=\frac{\mathrm{iii}}{5}.$ Since the equation $\frac{60}{100}=\frac{\mathrm{iii}}{\mathrm{five}}$ shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:

$$\frac{\text{amount}}{\text{base of operations}}=\frac{\text{pct}}{100}$$

$$\frac{\mathrm{iii}}{5}=\frac{60}{100}$$

Pct Proportion

The amount is to the base every bit the percentage is to $100.$

$$\frac{\text{amount}}{\text{base}}=\frac{\text{percent}}{100}$$

If nosotros restate the problem in the words of a proportion, it may be easier to set the proportion:

$$\mathit{\text{The amount is to the base as the percent is to 1 hundred.}}$$

We could also say:

$$\mathit{\text{The amount out of the base is the same as the percent out of one hundred.}}$$

First we volition practise translating into a percent proportion. Later, we'll solve the proportion.

Example half dozen.48

Translate to a proportion. What number is $\text{75\%}$ of $\mathrm{ninety}?$

Attempt It six.95

Translate to a proportion: What number is $\text{60\%}$ of $105?$

Effort It 6.96

Translate to a proportion: What number is $\text{40\%}$ of $85?$

Instance vi.49

Translate to a proportion. $19$ is $\text{25\%}$ of what number?

Try It 6.97

Translate to a proportion: $36$ is $\text{25\%}$ of what number?

Try It 6.98

Translate to a proportion: $27$ is $\text{36\%}$ of what number?

Example six.fifty

Interpret to a proportion. What percent of $27$ is $\mathrm{nine}?$

Try It 6.99

Translate to a proportion: What pct of $52$ is $39?$

Try It half dozen.100

Translate to a proportion: What percent of $92$ is $23?$

Interpret and Solve Percent Proportions

Now that nosotros accept written per centum equations as proportions, we are ready to solve the equations.

Case 6.51

Interpret and solve using proportions: What number is $\text{45\%}$ of $80?$

Try It six.101

Translate and solve using proportions: What number is $\text{65\%}$ of $40?$

Effort It 6.102

Translate and solve using proportions: What number is $\text{85\%}$ of $40?$

In the adjacent example, the pct is more than than $100,$ which is more 1 whole. So the unknown number will be more than the base.

Case 6.52

Interpret and solve using proportions: $\text{125\%}$ of $25$ is what number?

Try It 6.103

Translate and solve using proportions: $\text{125\%}$ of $64$ is what number?

Try Information technology 6.104

Translate and solve using proportions: $\text{175\%}$ of $84$ is what number?

Percents with decimals and money are also used in proportions.

Instance 6.53

Interpret and solve: $\text{6.5\%}$ of what number is $\text{\$1.56}?$

Try It half-dozen.105

Translate and solve using proportions: $\text{viii.5\%}$ of what number is $\text{\$3.23}?$

Try Information technology vi.106

Translate and solve using proportions: $\text{seven.25\%}$ of what number is $\text{\$four.64}?$

Example 6.54

Translate and solve using proportions: What pct of $72$ is $9?$

Try Information technology 6.107

Interpret and solve using proportions: What percentage of $72$ is $27?$

Endeavor It 6.108

Interpret and solve using proportions: What percent of $92$ is $23?$

Section 6.five Exercises

Practise Makes Perfect

Utilize the Definition of Proportion

In the post-obit exercises, write each sentence as a proportion.

244 .

$\mathrm{iv}$ is to $\mathrm{xv}$ as $36$ is to $135.$

245 .

$7$ is to $\mathrm{nine}$ every bit $35$ is to $45.$

246 .

$12$ is to $\mathrm{v}$ as $96$ is to $40.$

247 .

$15$ is to $8$ every bit $75$ is to $40.$

248 .

$5$ wins in $7$ games is the same as $115$ wins in $161$ games.

249 .

$4$ wins in $\mathrm{ix}$ games is the same equally $36$ wins in $81$ games.

250 .

$8$ campers to $\mathrm{one}$ advisor is the same as $48$ campers to $\mathrm{six}$ counselors.

251 .

$6$ campers to $1$ counselor is the same every bit $48$ campers to $8$ counselors.

252 .

$\text{\$9.36}$ for $18$ ounces is the same equally $\text{\$ii.threescore}$ for $5$ ounces.

253 .

$\text{\$3.92}$ for $\mathrm{viii}$ ounces is the same equally $\text{\$one.47}$ for $\mathrm{three}$ ounces.

254 .

$\text{\$18.04}$ for $11$ pounds is the same as $\text{\$4.92}$ for $\mathrm{three}$ pounds.

255 .

$\text{\$12.42}$ for $27$ pounds is the aforementioned as $\text{\$v.52}$ for $12$ pounds.

In the following exercises, decide whether each equation is a proportion.

256 .

$\frac{7}{\mathrm{xv}}=\frac{56}{120}$

257 .

$\frac{5}{12}=\frac{45}{108}$

258 .

$\frac{\mathrm{xi}}{\mathrm{half\; dozen}}=\frac{21}{\mathrm{xvi}}$

259 .

$\frac{9}{\mathrm{four}}=\frac{39}{34}$

260 .

$\frac{12}{18}=\frac{4.99}{\mathrm{vii.56}}$

261 .

$\frac{9}{16}=\frac{2.16}{3.89}$

262 .

$\frac{13.5}{\mathrm{viii.five}}=\frac{31.05}{19.55}$

263 .

$\frac{\mathrm{10.ane}}{\mathrm{viii.4}}=\frac{\mathrm{iii.03}}{\mathrm{ii.52}}$

Solve Proportions

In the post-obit exercises, solve each proportion.

264 .

$\frac{\mathrm{10}}{56}=\frac{7}{\mathrm{eight}}$

266 .

$\frac{49}{63}=\frac{z}{9}$

267 .

$\frac{56}{72}=\frac{y}{9}$

268 .

$\frac{5}{a}=\frac{65}{117}$

269 .

$\frac{\mathrm{iv}}{b}=\frac{64}{144}$

270 .

$\frac{98}{154}=\frac{\mathrm{-7}}{p}$

271 .

$\frac{72}{156}=\frac{\mathrm{-half\; dozen}}{q}$

272 .

$\frac{a}{\mathrm{-8}}=\frac{\mathrm{-42}}{48}$

274 .

$\frac{\mathrm{ii.half-dozen}}{3.9}=\frac{c}{\mathrm{iii}}$

276 .

$\frac{\mathrm{ii.7}}{j}=\frac{\mathrm{0.nine}}{0.2}$

277 .

$\frac{2.8}{\mathrm{thou}}=\frac{\mathrm{2.i}}{\mathrm{1.five}}$

278 .

$\frac{\frac{1}{2}}{\mathrm{one}}=\frac{m}{8}$

279 .

$\frac{\frac{1}{\mathrm{three}}}{3}=\frac{9}{n}$

Solve Applications Using Proportions

In the following exercises, solve the proportion problem.

280 .

Pediatricians prescribe $5$ milliliters (ml) of acetaminophen for every $25$ pounds of a child's weight. How many milliliters of acetaminophen volition the doctor prescribe for Jocelyn, who weighs $45$ pounds?

281 .

Brianna, who weighs $6$ kg, just received her shots and needs a pain killer. The hurting killer is prescribed for children at $15$ milligrams (mg) for every $1$ kilogram (kg) of the child's weight. How many milligrams will the doctor prescribe?

282 .

At the gym, Carol takes her pulse for $10$ sec and counts $19$ beats. How many beats per minute is this? Has Ballad met her target heart rate of $140$ beats per minute?

283 .

Kevin wants to keep his center charge per unit at $160$ beats per minute while training. During his conditioning he counts $27$ beats in $10$ seconds. How many beats per minute is this? Has Kevin met his target heart rate?

284 .

A new energy drink advertises $106$ calories for $8$ ounces. How many calories are in $12$ ounces of the drink?

285 .

1 $12$ ounce tin can of soda has $150$ calories. If Josiah drinks the big $32$ ounce size from the local mini-mart, how many calories does he get?

286 .

Karen eats $\frac{1}{2}$ cup of oatmeal that counts for $\mathrm{ii}$ points on her weight loss programme. Her husband, Joe, tin can accept $\mathrm{three}$ points of oatmeal for breakfast. How much oatmeal tin can he have?

287 .

An oatmeal cookie recipe calls for $\frac{\mathrm{one}}{\mathrm{two}}$ loving cup of butter to make $4$ dozen cookies. Hilda needs to make $10$ dozen cookies for the bake auction. How many cups of butter volition she need?

288 .

Janice is traveling to Canada and will change $\text{\$250}$ U.s. dollars into Canadian dollars. At the electric current substitution rate, $\text{\$i}$ United states is equal to $\text{\$1.01}$ Canadian. How many Canadian dollars will she get for her trip?

289 .

Todd is traveling to United mexican states and needs to commutation $\text{\$450}$ into Mexican pesos. If each dollar is worth $12.29$ pesos, how many pesos volition he get for his trip?

290 .

Steve changed $\text{\$600}$ into $480$ Euros. How many Euros did he receive per US dollar?

291 .

Martha inverse $\text{\$350}$ Us into $385$ Australian dollars. How many Australian dollars did she receive per US dollar?

292 .

At the laundromat, Lucy inverse $\text{\$12.00}$ into quarters. How many quarters did she become?

293 .

When she arrived at a casino, Gerty inverse $\text{\$20}$ into nickels. How many nickels did she get?

294 .

Jesse's car gets $\mathrm{thirty}$ miles per gallon of gas. If Las Vegas is $285$ miles abroad, how many gallons of gas are needed to get there and then abode? If gas is $\text{\$iii.09}$ per gallon, what is the total cost of the gas for the trip?

295 .

Danny wants to drive to Phoenix to run across his grandfather. Phoenix is $370$ miles from Danny's domicile and his car gets $\mathrm{xviii.five}$ miles per gallon. How many gallons of gas will Danny need to get to and from Phoenix? If gas is $\text{\$three.19}$ per gallon, what is the total cost for the gas to drive to see his grandad?

296 .

Hugh leaves early on ane forenoon to drive from his home in Chicago to go to Mount Rushmore, $812$ miles away. Subsequently $3$ hours, he has gone $190$ miles. At that rate, how long will the whole drive take?

297 .

Kelly leaves her domicile in Seattle to drive to Spokane, a distance of $280$ miles. Subsequently $2$ hours, she has gone $152$ miles. At that rate, how long will the whole drive take?

298 .

Phil wants to fertilize his lawn. Each bag of fertilizer covers well-nigh $\mathrm{4,000}$ square feet of lawn. Phil's lawn is approximately $\mathrm{13,500}$ square feet. How many bags of fertilizer volition he have to buy?

299 .

April wants to paint the exterior of her house. One gallon of pigment covers about $350$ foursquare feet, and the exterior of the business firm measures approximately $2000$ foursquare feet. How many gallons of paint volition she take to buy?

Write Per centum Equations as Proportions

In the following exercises, translate to a proportion.

300 .

What number is $\text{35\%}$ of $250?$

301 .

What number is $\text{75\%}$ of $920?$

302 .

What number is $\text{110\%}$ of $47?$

303 .

What number is $\text{150\%}$ of $64?$

304 .

$45$ is $\text{xxx\%}$ of what number?

305 .

$25$ is $\text{80\%}$ of what number?

306 .

$90$ is $\text{150\%}$ of what number?

307 .

$77$ is $\text{110\%}$ of what number?

308 .

What percent of $85$ is $17?$

309 .

What percent of $92$ is $46?$

310 .

What percentage of $260$ is $340?$

311 .

What pct of $180$ is $220?$

Translate and Solve Percent Proportions

In the following exercises, translate and solve using proportions.

312 .

What number is $\text{65\%}$ of $180?$

313 .

What number is $\text{55\%}$ of $300?$

314 .

$\text{18\%}$ of $92$ is what number?

315 .

$\text{22\%}$ of $74$ is what number?

316 .

$\text{175\%}$ of $26$ is what number?

317 .

$\text{250\%}$ of $61$ is what number?

318 .

What is $\text{300\%}$ of $488?$

319 .

What is $\text{500\%}$ of $315?$

320 .

$\text{17\%}$ of what number is $\text{\$7.65}?$

321 .

$\text{19\%}$ of what number is $\text{\$half dozen.46}?$

322 .

$\text{\$13.53}$ is $\text{8.25\%}$ of what number?

323 .

$\text{\$eighteen.12}$ is $\text{7.55\%}$ of what number?

324 .

What percentage of $56$ is $14?$

325 .

What percent of $80$ is $28?$

326 .

What per centum of $96$ is $12?$

327 .

What percent of $120$ is $27?$

Everyday Math

328 .

Mixing a concentrate Sam bought a large canteen of full-bodied cleaning solution at the warehouse shop. He must mix the concentrate with water to make a solution for washing his windows. The directions tell him to mix $\mathrm{three}$ ounces of concentrate with $5$ ounces of h2o. If he puts $12$ ounces of concentrate in a bucket, how many ounces of h2o should he add? How many ounces of the solution will he take birthday?

329 .

Mixing a concentrate Travis is going to wash his car. The directions on the canteen of automobile launder concentrate say to mix $2$ ounces of concentrate with $15$ ounces of water. If Travis puts $\mathrm{six}$ ounces of concentrate in a bucket, how much water must he mix with the concentrate?

Writing Exercises

330 .

To solve "what number is $\text{45\%}$ of $350\text{"}$ practise you adopt to utilize an equation similar you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

331 .

To solve "what percent of $125$ is $25\text{"}$ do you lot prefer to use an equation like y'all did in the department on Decimal Operations or a proportion similar you did in this section? Explicate your reason.

Self Cheque

ⓐ Later on completing the exercises, use this checklist to evaluate your mastery of the objectives of this department.

ⓑ Overall, after looking at the checklist, do you lot retrieve y'all are well-prepared for the adjacent Chapter? Why or why non?

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Source: https://openstax.org/books/prealgebra/pages/6-5-solve-proportions-and-their-applications

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